Quadratic BSDEs with jumps and related PIDEs

In this paper we are interested to solve a class of quadratic BSDEs with jumps (QBSDEJs for short) of the following form: Y-t = xi + integral(T)(t) H(Y-s, Z(s), U-s (.)) ds - integral(T)(t) Z(s)dW(s )- integral t(T) integral E U-s(e)(N) over tilde (ds, de), Herein, the terminal data xi will be assumed to be square integrable. Our study covers the following cases H(y, z, u(.))= {f(y) vertical bar z vertical bar(2) + [u](f)(y) =: H-f(y, z, u(.)) h(y, u(.)) + cz + H-f(y, z, u(.))) a + b vertical bar y vertical bar + c vertical bar z vertical bar + d parallel to u(.)parallel to(nu,1) + H-f(y, z, u(.)) cz + f(y) vertical bar z vertical bar(2) - integral(E) u(e)nu(de) cz + f(y) vertical bar z vertical bar(2) h (y, u(.)) + cz + f(y) vertical bar z vertical bar(2) H-0 (r, X-r) + H-f (y, z, u(.))), (X-r)(r >= 0) is a markov process where f is a measurable and integrable function, [u](f) (.) is a functional of the unknown processes Y and U. (.) to be defined later and h and Ho enjoy some classical assumptions. The generators show quadratic growth in the Brownian component and non-linear functional form with respect to the jump term. Existence or uniqueness of solutions as well as a comparison and strict comparison principles are established under no monotonicity condition in the third argument of the generator. Probabilistic representations of solutions to some classes of quadratic PIDE are given by means of solutions of these QBSDEJs. The main idea is to use a phase space transformation to transform our initial QBSDEJ to a standard BSDEJ without quadratic term.